(To be removed) Simulate univariate GARCH(P,Q) process with Gaussian innovations
As an alternative to ugarchsim
, use
the garch
object
to create conditional variance models. For more information, see Specify GARCH Models Using garch.
ugarchsim
is removed. Use garch
object to create conditional
variance models and the simulate
function
to generate Monte Carlo simulations from conditional variance models.
For more information, see Simulate Conditional Variance Model.
[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)
 Scalar constant term [[KAPPA]] of the GARCH process. 




 Positive, scalar integer indicating the number of samples
of the innovations 
[U, H] = ugarchsim(Kappa, Alpha, Beta, NumSamples)
simulates
a univariate GARCH(P,Q) process with Gaussian innovations.
U
is a number of samples (NUMSAMPLES
)by1
vector
of innovations (ɛ_{t}),
representing a meanzero, discretetime stochastic process. The innovations
time series U
is designed to follow the GARCH(P,Q)
process specified by the inputs Kappa
, Alpha
,
and Beta
.
H
is a NUMSAMPLES
by1
vector
of the conditional variances (
) corresponding to the innovations
vector U
. Note that U
and H
are
the same length, and form a "matching" pair of vectors.
As shown in the following equation, $${\sigma}_{t}^{2}$$ (that
is, H(t)
) represents the time series inferred from
the innovations time series {ɛ_{t}}
(that is, U
).
The timeconditional variance, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{ti}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{tj}^{2}},$$
where α represents the argument Alpha
,
β represents Beta
, and the GARCH(P,Q) coefficients
{Κ, α, β}
are subject to the following constraints.
$$\begin{array}{l}{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}}<1\\ K>0\\ \begin{array}{cc}{\alpha}_{i}\ge 0& i=1,2,\dots ,P\\ {\beta}_{j}\ge 0& j=1,2,\dots ,Q.\end{array}\end{array}$$
Note that U
is a vector of residuals or innovations
(ɛ_{t})
of an econometric model, representing a meanzero, discretetime stochastic
process.
Although $${\sigma}_{t}^{2}$$ is generated using the equation above, ɛ_{t} and $${\sigma}_{t}^{2}$$ are related as
$${\epsilon}_{t}={\sigma}_{t}{\upsilon}_{t},$$
where $$\left\{{\upsilon}_{t}\right\}$$ is an independent, identically distributed (iid) sequence ~ N(0,1).
The output vectors U
and H
are
designed to be steadystate sequences in which transients have arbitrarily
small effect. The (arbitrary) metric used by ugarchsim
strips
the first N
samples of U
and H
such
that the sum of the GARCH coefficients, excluding Kappa
,
raised to the N
th power, does not exceed 0.01.
0.01 = (sum(Alpha) + sum(Beta))^N
Thus
N = log(0.01)/log((sum(Alpha) + sum(Beta)))
Note
The Econometrics Toolbox™ software provides a comprehensive
and integrated computing environment for the analysis of volatility
in time series. For information see the Econometrics Toolbox documentation
or the financial products Web page at 
This example simulates a GARCH(P,Q) process with P
= 2
and Q = 1
.
% Set the random number generator seed for reproducability. rng('default') % Set the simulation parameters of GARCH(P,Q) = GARCH(2,1) process. Kappa = 0.25; %a positive scalar. Alpha = [0.2 0.1]'; %a column vector of nonnegative numbers (P = 2). Beta = 0.4; % Q = 1. NumSamples = 500; % number of samples to simulate. % Now simulate the process. [U , H] = ugarchsim(Kappa, Alpha, Beta, NumSamples); % Estimate the process parameters. P = 2; % Model order P (P = length of Alpha). Q = 1; % Model order Q (Q = length of Beta). [k, a, b] = ugarch(U , P , Q); disp(' ') disp(' Estimated Coefficients:') disp(' ') disp([k; a; b]) disp(' ') % Forecast the conditional variance using the estimated % coefficients. NumPeriods = 10; % Forecast out to 10 periods. [VarianceForecast, H1] = ugarchpred(U, k, a, b, NumPeriods); disp(' Variance Forecasts:') disp(' ') disp(VarianceForecast) disp(' ')
Error using ugarchsim (line 71) UGARCHSIM has been removed. Use the GARCH model SIMULATE function in Econometrics Toolbox instead.
James D. Hamilton, Time Series Analysis, Princeton University Press, 1994